2. The ergodicity problem in economics¶

Q 2.3 Coding the problem¶

1) Generate a pandas dataframe that simulates N coin tosses for w gamblers with the np.random.rand function. Set N=100 coin tosses and w=100 gamblers and add the prefix ’prob ’ to the column name.

In [1]:
import pandas as pd
import numpy as np

n = 100
w = 100

df = pd.DataFrame()
for i in range(100):
    data = np.random.randint(0, 2, size=n)
    temp_df = pd.DataFrame(data, columns=[f'prob_{i}'])
    df = pd.concat([df, temp_df], axis=1)

df
Out[1]:
prob_0 prob_1 prob_2 prob_3 prob_4 prob_5 prob_6 prob_7 prob_8 prob_9 ... prob_90 prob_91 prob_92 prob_93 prob_94 prob_95 prob_96 prob_97 prob_98 prob_99
0 0 1 0 0 0 1 0 1 0 1 ... 0 0 0 1 1 1 1 1 1 1
1 1 1 0 0 1 1 0 1 1 1 ... 1 1 0 1 0 0 0 1 1 1
2 1 0 1 1 1 1 0 0 1 0 ... 1 1 0 0 1 0 0 1 0 0
3 1 1 1 0 1 1 1 1 1 1 ... 1 1 1 1 1 0 1 0 0 1
4 0 0 1 1 0 0 1 1 0 0 ... 1 1 1 0 0 0 1 1 1 1
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
95 0 0 1 0 1 0 1 1 0 0 ... 1 1 1 0 0 0 1 0 1 0
96 1 1 1 1 1 0 0 1 1 1 ... 0 0 1 0 0 0 0 0 1 1
97 1 0 1 1 1 0 0 0 0 0 ... 1 0 0 0 0 0 1 1 0 1
98 1 1 1 1 0 1 1 0 0 1 ... 1 0 0 0 1 1 1 1 1 1
99 0 1 1 0 0 0 0 1 1 0 ... 1 1 1 0 0 0 0 1 1 0

100 rows × 100 columns

2) Initialize an additional w columns [’balance ’ + str(x)] with the value 100 to capture the initial wealth of 100$, where: x in range(0, w).

In [2]:
for x in range(0, w):
    data = [w] * n
    temp_df = pd.DataFrame(data, columns=[f'balance {x}'])
    df = pd.concat([df, temp_df], axis=1)

df
Out[2]:
prob_0 prob_1 prob_2 prob_3 prob_4 prob_5 prob_6 prob_7 prob_8 prob_9 ... balance 90 balance 91 balance 92 balance 93 balance 94 balance 95 balance 96 balance 97 balance 98 balance 99
0 0 1 0 0 0 1 0 1 0 1 ... 100 100 100 100 100 100 100 100 100 100
1 1 1 0 0 1 1 0 1 1 1 ... 100 100 100 100 100 100 100 100 100 100
2 1 0 1 1 1 1 0 0 1 0 ... 100 100 100 100 100 100 100 100 100 100
3 1 1 1 0 1 1 1 1 1 1 ... 100 100 100 100 100 100 100 100 100 100
4 0 0 1 1 0 0 1 1 0 0 ... 100 100 100 100 100 100 100 100 100 100
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
95 0 0 1 0 1 0 1 1 0 0 ... 100 100 100 100 100 100 100 100 100 100
96 1 1 1 1 1 0 0 1 1 1 ... 100 100 100 100 100 100 100 100 100 100
97 1 0 1 1 1 0 0 0 0 0 ... 100 100 100 100 100 100 100 100 100 100
98 1 1 1 1 0 1 1 0 0 1 ... 100 100 100 100 100 100 100 100 100 100
99 0 1 1 0 0 0 0 1 1 0 ... 100 100 100 100 100 100 100 100 100 100

100 rows × 200 columns

3) Create a two-level for loop that iterates through rows 1 to N first and columns w to 2×w next implementing the calculation logic for every prob-balance pair:

Screenshot%202022-10-14%20at%2010.06.18%20PM.png

In [3]:
for i in range(1,n):
    for j in range(w, 2*w):
        prob_value = df.iloc[i-1,j-100]
        if prob_value == 1:
            df.iloc[i,j] = df.iloc[i-1,j] * 1.5
        else:
            df.iloc[i,j] = df.iloc[i-1,j] * 0.6

df
Out[3]:
prob_0 prob_1 prob_2 prob_3 prob_4 prob_5 prob_6 prob_7 prob_8 prob_9 ... balance 90 balance 91 balance 92 balance 93 balance 94 balance 95 balance 96 balance 97 balance 98 balance 99
0 0 1 0 0 0 1 0 1 0 1 ... 100.000000 100.000000 100.000000 100.000000 100.000000 100.000000 100.000000 100.000000 100.000000 100.000000
1 1 1 0 0 1 1 0 1 1 1 ... 60.000000 60.000000 60.000000 150.000000 150.000000 150.000000 150.000000 150.000000 150.000000 150.000000
2 1 0 1 1 1 1 0 0 1 0 ... 90.000000 90.000000 36.000000 225.000000 90.000000 90.000000 90.000000 225.000000 225.000000 225.000000
3 1 1 1 0 1 1 1 1 1 1 ... 135.000000 135.000000 21.600000 135.000000 135.000000 54.000000 54.000000 337.500000 135.000000 135.000000
4 0 0 1 1 0 0 1 1 0 0 ... 202.500000 202.500000 32.400000 202.500000 202.500000 32.400000 81.000000 202.500000 81.000000 202.500000
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
95 0 0 1 0 1 0 1 1 0 0 ... 1618.114584 0.010859 647.245833 2.651119 2.651119 0.001737 647.245833 6.627797 258.898333 103.559333
96 1 1 1 1 1 0 0 1 1 1 ... 2427.171875 0.016288 970.868750 1.590671 1.590671 0.001042 970.868750 3.976678 388.347500 62.135600
97 1 0 1 1 1 0 0 0 0 0 ... 1456.303125 0.009773 1456.303125 0.954403 0.954403 0.000625 582.521250 2.386007 582.521250 93.203400
98 1 1 1 1 0 1 1 0 0 1 ... 2184.454688 0.005864 873.781875 0.572642 0.572642 0.000375 873.781875 3.579011 349.512750 139.805100
99 0 1 1 0 0 0 0 1 1 0 ... 3276.682032 0.003518 524.269125 0.343585 0.858963 0.000563 1310.672813 5.368516 524.269125 209.707650

100 rows × 200 columns

In [4]:
df.to_csv('df.csv')

4) Plot the time trajectories (in gamble iterations) of the wealth per gambler and calculate how many gamblers have more than their initial 100 dollars. What happens when you increase N to 1,000 (Figure 2 in the original paper, except the red and blue line)? How many gamblers with more than 100 dollars do you have at t=1,000?

In [5]:
count = 0

for i in range(100,200):
    if df.iloc[-1,i] > 100:
        count += 1

print(f'Number of gambler whose wealth is more than 100$ : {count}')

Number of gambler whose wealth is more than 100$ : 18
In [6]:
import matplotlib.pyplot as plt

for i in range(100,200):
    temp_df = df.iloc[:,[i]].copy()
    plt.plot(temp_df)
    plt.title(df.columns[i])
    plt.xlabel('Time in gamble rounds')
    plt.ylabel('Wealth')
    plt.show()
In [7]:
plt.figure(figsize=(12, 12))
plt.xlabel('Time in gamble rounds')
plt.ylabel('Wealth')
display(plt.plot(df))

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N = 1000¶

In [8]:
n = 1000
w = 100

df1 = pd.DataFrame()
for i in range(100):
    data = np.random.randint(0, 2, size=n)
    temp_df = pd.DataFrame(data, columns=[f'prob_{i}'])
    df1 = pd.concat([df1, temp_df], axis=1)

for x in range(0, w):
    data = [w] * n
    temp_df = pd.DataFrame(data, columns=[f'balance {x}'])
    df1 = pd.concat([df1, temp_df], axis=1)

for i in range(1,n):
    for j in range(w, 2*w):
        prob_value = df1.iloc[i-1,j-100]
        if prob_value == 1:
            df1.iloc[i,j] = df1.iloc[i-1,j] * 1.5
        else:
            df1.iloc[i,j] = df1.iloc[i-1,j] * 0.6

df1.to_csv('df1.csv')

count = 0

for i in range(100,200):
    if df1.iloc[-1,i] > 100:
        count += 1

print(f'Number of gambler whose wealth is more than 100$ : {count}')

Number of gambler whose wealth is more than 100$ : 0
In [9]:
for i in range(100,200):
    temp_df = df1.iloc[:,[i]].copy()
    plt.plot(temp_df)
    plt.title(df1.columns[i])
    plt.xlabel('Time in gamble rounds')
    plt.ylabel('Wealth')
    plt.show()
In [10]:
plt.figure(figsize=(24, 24))
plt.xlabel('Time in gamble rounds')
plt.ylabel('Wealth')
display(plt.plot(df1))

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5) Plot the time trajectories of the mean and median wealth at every time point from 0 to 1,000. Try both linear and logarithmic scaler for the wealth axis. What are your observations?

In [11]:
mean_median_df = df1.copy()
w = 100

df1['mean'] = df1.iloc[:, w:2*w].mean(axis=1)
df1['median'] = df1.iloc[:, w:2*w].median(axis=1)
df1
Out[11]:
prob_0 prob_1 prob_2 prob_3 prob_4 prob_5 prob_6 prob_7 prob_8 prob_9 ... balance 92 balance 93 balance 94 balance 95 balance 96 balance 97 balance 98 balance 99 mean median
0 1 0 1 1 0 1 0 1 1 0 ... 1.000000e+02 1.000000e+02 1.000000e+02 1.000000e+02 1.000000e+02 1.000000e+02 1.000000e+02 1.000000e+02 1.000000e+02 1.000000e+02
1 0 1 0 0 0 1 0 0 0 1 ... 6.000000e+01 6.000000e+01 1.500000e+02 6.000000e+01 1.500000e+02 1.500000e+02 6.000000e+01 1.500000e+02 1.167000e+02 1.500000e+02
2 1 1 0 1 0 1 1 1 0 0 ... 3.600000e+01 9.000000e+01 9.000000e+01 3.600000e+01 9.000000e+01 9.000000e+01 9.000000e+01 9.000000e+01 1.116000e+02 9.000000e+01
3 0 0 0 0 1 0 0 0 1 1 ... 5.400000e+01 5.400000e+01 1.350000e+02 5.400000e+01 1.350000e+02 5.400000e+01 5.400000e+01 1.350000e+02 1.230120e+02 1.350000e+02
4 0 1 0 0 1 0 1 1 1 1 ... 3.240000e+01 3.240000e+01 8.100000e+01 8.100000e+01 2.025000e+02 3.240000e+01 8.100000e+01 8.100000e+01 1.225287e+02 8.100000e+01
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
995 0 0 0 0 1 0 0 1 0 1 ... 1.114239e-23 5.018087e-34 2.720310e-21 6.333716e-15 6.485725e-17 2.533486e-15 7.131130e-25 1.037716e-17 2.070404e-08 4.760543e-21
996 0 1 1 0 0 1 0 1 1 1 ... 1.671359e-23 3.010852e-34 4.080465e-21 3.800230e-15 9.728588e-17 1.520092e-15 4.278678e-25 1.556574e-17 1.793922e-08 4.080465e-21
997 0 0 0 0 0 1 1 1 0 0 ... 2.507038e-23 1.806511e-34 6.120698e-21 5.700344e-15 1.459288e-16 2.280138e-15 6.418017e-25 2.334861e-17 1.862089e-08 4.284489e-21
998 1 0 0 1 0 0 0 0 1 1 ... 3.760557e-23 2.709767e-34 9.181047e-21 8.550517e-15 8.755729e-17 1.368083e-15 9.627026e-25 1.400917e-17 1.626547e-08 6.426733e-21
999 1 0 1 1 1 1 0 1 0 1 ... 2.256334e-23 4.064650e-34 1.377157e-20 5.130310e-15 1.313359e-16 2.052124e-15 1.444054e-24 8.405500e-18 9.938575e-09 5.508628e-21

1000 rows × 202 columns

In [12]:
df1['mean'].plot()
plt.xlabel('Time in gamble rounds')
plt.ylabel('Wealth')
plt.show()
In [13]:
df1['median'].plot()
plt.xlabel('Time in gamble rounds')
plt.ylabel('Wealth')
plt.show()

6) Repeat steps 3-5 with N=10 and w=1,000 with the following logic: image.png

In [14]:
n = 10
w = 1000

df2 = pd.DataFrame()
for i in range(1000):
    data = np.random.randint(0, 2, size=n)
    temp_df = pd.DataFrame(data, columns=[f'prob_{i}'])
    df2 = pd.concat([df2, temp_df], axis=1)

for x in range(0, w):
    data = [100] * n
    temp_df = pd.DataFrame(data, columns=[f'balance {x}'])
    df2 = pd.concat([df2, temp_df], axis=1)

for i in range(1,n):
    for j in range(w, 2*w):
        prob_value = df2.iloc[i-1,j-1000]
        if prob_value == 1:
            df2.iloc[i,j] = df2.iloc[i-1,j] + 50
        else:
            df2.iloc[i,j] = df2.iloc[i-1,j] - 40

df2.to_csv('df2.csv')
df2
Out[14]:
prob_0 prob_1 prob_2 prob_3 prob_4 prob_5 prob_6 prob_7 prob_8 prob_9 ... balance 990 balance 991 balance 992 balance 993 balance 994 balance 995 balance 996 balance 997 balance 998 balance 999
0 0 1 1 0 0 1 0 0 0 1 ... 100 100 100 100 100 100 100 100 100 100
1 0 0 1 0 0 1 0 0 0 0 ... 60 150 150 150 60 150 150 150 150 60
2 0 0 0 0 0 1 0 1 0 0 ... 110 110 200 200 110 200 110 200 110 20
3 1 0 0 1 0 0 1 1 0 1 ... 160 70 250 160 160 160 160 250 70 -20
4 0 1 1 1 0 1 0 0 0 1 ... 120 30 210 210 120 120 120 210 30 -60
5 0 1 1 0 0 1 0 1 0 0 ... 170 80 260 170 170 80 170 170 80 -10
6 1 0 1 0 1 0 0 1 0 1 ... 220 130 220 130 220 130 130 220 40 -50
7 1 1 0 0 1 1 0 1 1 1 ... 270 90 180 90 270 90 90 270 90 -90
8 1 0 1 0 0 1 1 1 1 0 ... 320 140 140 140 320 50 140 230 140 -130
9 1 1 0 1 1 1 1 0 0 0 ... 280 100 190 100 280 100 100 190 100 -80

10 rows × 2000 columns

In [15]:
count = 0

for i in range(1000,2000):
    if df2.iloc[-1,i] > 1000:
        count += 1

print(f'Number of gambler whose wealth is more than $1000 : {count}')

Number of gambler whose wealth is more than $1000 : 0
In [16]:
for i in range(100,200):
    temp_df = df2.iloc[:,[i]].copy()
    plt.plot(temp_df)
    plt.title(df2.columns[i])
    plt.xlabel('Time in gamble rounds')
    plt.ylabel('Wealth')
    plt.show()
In [17]:
plt.figure(figsize=(12, 12))
plt.xlabel('Time in gamble rounds')
plt.ylabel('Wealth')
display(plt.plot(df2))

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In [18]:
mean_median_df = df2.copy()
w = 1000

df2['mean'] = df2.iloc[:, w:2*w].mean(axis=1)
df2['median'] = df2.iloc[:, w:2*w].median(axis=1)
df2
Out[18]:
prob_0 prob_1 prob_2 prob_3 prob_4 prob_5 prob_6 prob_7 prob_8 prob_9 ... balance 992 balance 993 balance 994 balance 995 balance 996 balance 997 balance 998 balance 999 mean median
0 0 1 1 0 0 1 0 0 0 1 ... 100 100 100 100 100 100 100 100 100.00 100.0
1 0 0 1 0 0 1 0 0 0 0 ... 150 150 60 150 150 150 150 60 106.17 150.0
2 0 0 0 0 0 1 0 1 0 0 ... 200 200 110 200 110 200 110 20 112.43 110.0
3 1 0 0 1 0 0 1 1 0 1 ... 250 160 160 160 160 250 70 -20 118.06 160.0
4 0 1 1 1 0 1 0 0 0 1 ... 210 210 120 120 120 210 30 -60 121.62 120.0
5 0 1 1 0 0 1 0 1 0 0 ... 260 170 170 80 170 170 80 -10 124.91 170.0
6 1 0 1 0 1 0 0 1 0 1 ... 220 130 220 130 130 220 40 -50 129.46 130.0
7 1 1 0 0 1 1 0 1 1 1 ... 180 90 270 90 90 270 90 -90 134.37 180.0
8 1 0 1 0 0 1 1 1 1 0 ... 140 140 320 50 140 230 140 -130 141.62 140.0
9 1 1 0 1 1 1 1 0 0 0 ... 190 100 280 100 100 190 100 -80 149.32 190.0

10 rows × 2002 columns

In [19]:
df2['mean'].plot()
plt.xlabel('Time in gamble rounds')
plt.ylabel('Wealth')
plt.show()
In [20]:
df2['median'].plot()
plt.xlabel('Time in gamble rounds')
plt.ylabel('Wealth')
plt.show()

These new results are more favorable to the gambler and as they actually get to leave the place with more money than before, whereas the previous models indicate that after a certain number of rounds, the money eventually leaves the gamblers pocket and enters the casino, therefore making it a loss filled deal and gamble.

In [ ]: